$\begingroup$Usually, if we can get $< 2 \epsilon$, we say GOOD ENOUGH! since, in most cases, we could have just said "let such and such so that it is less than $\epsilon / 2$" rather than "let such and such so that it is less than $\epsilon$."$\endgroup$
– MCTMar 24 '15 at 1:59

$\begingroup$@Soke That's why I like to imagine that I'm working with infinitesimals in calculus, rather than formal, epsilon-delta constructions. If we get it $<2\epsilon$, that's OK, because $2\epsilon$ is also an infinitesimal.$\endgroup$
– Akiva WeinbergerMar 24 '15 at 2:21

First of all you've made a mistake: you need to introduce $N_1$ and $N_2$ so that for any $m_1,n_1 \geq N_1$ you have the property and similar for the other one.

Having fixed that, if you have $|a_m + b_m - a_n - b_n| < 2 \varepsilon$ for $m,n \geq N$, then you are technically done, since $2 \varepsilon$ can be made arbitrarily small by making $\varepsilon$ arbitrarily small.

Typically for aesthetics, you let $\varepsilon$ be the small number for the converging quantity of interest and then choose new small numbers for what contributes to it from there. Here you can take $\varepsilon$ to be your small number for $a_n+b_n$ and then choose $N_1,N_2$ so that you get $\varepsilon/2$-closeness for $a_n$ and $b_n$ respectively. This is valid because the definition that $a_n$ and $b_n$ are Cauchy says you can pick whatever "$\varepsilon$" you want, so you can in particular pick it to be $\varepsilon/2$ (where $\varepsilon$ was already specified).

You can choose $\epsilon_1$ and $\epsilon_2$ to be whatever you want, so choose them to be $\frac{\epsilon}{2}$. Choosing a value for $\epsilon_1$ determines a value for $m_1$ and $n_1$, but you can start with any value for $\epsilon_1$ ( and similarly for $\epsilon_2$ ).

The statement that ${a_n}$ is Cauchy is that for any $\epsilon$ there exists an $M$ and $N$ such that for any $n>N$ and $m>M$, we have $|a_m - a_n| < \epsilon$. The quantifiers there are important: $M$ and $N$ depend on the choice for $\epsilon$. But once you know the sequence is Cauchy, you can choose any $\epsilon$, and that will give you a $M$ and $N$. So, typically in these proofs, when you're doing them, you start with what you've done, you get to the end and look at the bound you found ( $\epsilon_1 + \epsilon_2$ ), and then figure out what you needed to start with, for example that $\epsilon_1 < \frac{\epsilon}{2}$ and $\epsilon_2 < \frac{\epsilon}{2}$

You can let $\epsilon' = 2\epsilon$ to complete the proof. However, since in NA people (for whatever reason) insist on using $\epsilon$ at the end of proofs, you could also complete the proof by saying something like "Lastly, interchanging $\epsilon$ and $\epsilon'$ completes the proof."